Add animation to armature which is already animated
Most animation data recorded in blender gets saved as an action, an action can easily be applied to another object, it will work fine as long as the animated data path names match in both objects. For an armature that means an action created with one armature can be applied to another armature and it will move bones that have the same name as the first armature.

For non-matching armatures not all is lost, but could be more work than it's worth. You can add constraints to have a bone follow the motion of another bone. If your animation is simple it may be easier to just animate the new armature, you can use the original as a reference to make it easier.